It might be more accurate to ask, "Is there any situation where one would want to log in as the local administrator account on a domain-joined server, and not a domain account?"
I never log in as the local administrator account of a domain-joined server. I prefer to give it a complex password and then disable it, but my current colleagues prefer to give it a complex password and not use it. If you think you're going to lose AD connectivity and don't allow cached credentials, you might want to log in as the local administrator.
That's not the question you asked, though.
On the server that is connected to the domain controller, should I be
logging in as the local Administrator or a domain user?
Don't share accounts. Log in as a domain account that's specific to you.
Is there any reason or situation where one would want to log in as a
domain user on a server, and not the Admin account?
When more than one person manages the server and you want to keep track of who does what. When more than one person manages the server and you don't want your coworkers to change the password and forget to tell you what it is. When you're the only person who manages a server and don't want to boot the server off a Hiren's if you forget the administrator password.
When you want to use remote management tools. When you're going to need to map network drives and don't want to type usernames and passwords. When you're setting up or managing network services that assume domain credentials.
I'm sure there are more, but I'm going to stop there.